Now \(\theta _0\)  is the cohomological map associated to the Teichmuller map \(\sharp :_m^_m\) . Scholze showed in [1]} Proposition 5.20 that the Teichmuller map on the perfectoid Tate algebra maps \(T_i\mapsto T_i\) . View \(\theta _0\)  as a map on Čech cohomology with respect to the standard affine covers, and view \(^1(^{n,},_m)\)  as descent data for building a line bundle (and similarly for the tilt). Then we see that \(\sharp \)  sends descent data for \((1)\)  (which is monomials of degree one), to monomials of degree one, which build \((1)\)  on \(^{n,}_K\) .
